Splits in the Cabinet and the wider Conservative Party have been exposed by the ongoing row over post-Brexit customs arrangements.

Theresa May was warned that attempts to keep her preferred new “customs partnership” model alive are “desperate” after it was rejected by the Brexit “war cabinet” of senior ministers.

She also faced pressure from pro-EU Tories to crack down on Jacob Rees-Mogg’s European Research Group amid claims the Brexiteers were effectively holding the Prime Minister to ransom with their demands.

The divisions were laid bare after Business Secretary Greg Clark stressed that thousands of British jobs depend on frictionless trade with Europe, in what was viewed as an attempt to revive the customs partnership model.

Environment Secretary Michael Gove risked fuelling the row after describing as “helpful” a Twitter thread by former aide Henry Newman – now director of the Open Europe think tank – which described resurrecting the customs partnership as “surely misguided”.

The Business Secretary was part of the Brexit “war cabinet” which failed to reach agreement on whether to back the hybrid customs partnership – which would see the UK collect import duties on behalf of the EU for goods arriving via British ports and airports – or the so-called “maximum facilitation” or “max fac” model relying on the extensive use of technology to minimise checks at the border.

Brexiteers have rejected the customs partnership, claiming it would keep the UK too closely tied to Brussels and Mrs May’s inner war cabinet is thought to have lined up six-five against it, despite the Prime Minister and Mr Clark backing the model.

Mr Newman said if Number 10 sought to pursue a version of the customs partnership, “they will be putting all their eggs in a broken basket” and it was “far better to accept some friction on trade” and pursue the “max fac” model.

Both Mr Gove and Mr Clark were part of the Brexit Cabinet committee which discussed the customs options.

The Business Secretary denied being close to tears as he spoke at Wednesday’s meeting, but admitted “I do feel very strongly” about it, before highlighting the importance of having a customs deal with “the minimum of frictions” to firms operating a just-in-time production line such as Toyota, which employs 3,500 people at plants in North Wales and Derbyshire.

Mr Clark added: “You can compare models but actually you need to have in mind the future jobs but also the very important jobs for people today.”

The Business Secretary also opened the door to extending a transition period on customs with the EU until a new arrangement was ready.

His comments provoked fury from Brexiteers, with one Cabinet source telling the Press Association: “Greg appears to be re-fighting the referendum, this is Project Fear 3.0.

“Instead of listening to vested interests, he should pay more attention to the 17.4 million who voted to take back control of our trade policy.

“The Customs Partnership has been roundly rejected, making threats to keep it alive is just desperate.”

But Mr Clark was backed by former home secretary Amber Rudd, who said Mr Clark was right to argue the case “for a Brexit that protects existing jobs and future investment”.

Business groups the British Chambers of Commerce and the CBI also welcomed Mr Clark’s comments, saying it was important to maintain the status quo on frictionless trade until a new arrangement is in place.

The swift show of support for Mr Clark led to speculation the move had been orchestrated by Number 10, a claim dismissed as “nonsense” by a Downing Street insider.

Jacob Rees-Mogg, chairman of the influential European Research Group (ERG) of Tory MPs hit out at Mr Clark’s suggestion jobs could be at risk, saying: “This Project Fear has been so thoroughly discredited that you would have thought it would have come to an end by now.”

He told ITV’s Peston on Sunday: “The customs partnership is in a sense misnamed because it means single market as well as customs union and therefore we would not in effect be leaving the European Union.”

Former Tory leader Iain Duncan Smith, writing on the ConservativeHome website, said: “Despite the crucial Cabinet sub-committee on Brexit negotiations and strategy deciding last week that the scheme isn’t fit for purpose, some in Downing Street are, incredibly, now briefing out that after a few tweaks it can be presented again.”

But pro-EU former business minister Anna Soubry urged Mrs May to “see off these ideologues who are blighting our party and blighting the Brexit process”.

And Sir Nicholas Soames said “many of us want this holding to ransom of the PM” by the ERG “to end”.

Many of us want this holding to ransom of The PM and the pragmatic centre of the Party and Public opinion by ERG and it’s supporters including @Jacob_Rees_Mogg to end ,and for the best Brexit for U.K. to be negotiated #itsgottostop

DUP leader Arlene Foster, whose party props up Mrs May’s government in Westminster, hit out at Brussels over the EU’s stance on customs.

She told The Andrew Marr Show: “We believe that there are ways to deal with this, and indeed, back in August of last year, as you know, the Government put forward various proposals. We were disappointed there wasn’t the engagement from the European Union at that time.”